Cook was born in New Brunswick and lived there until his teenage years when his family moved to Florida. He earned a master's degree in history from Ohio University in Athens, OH in 1962 and was signed by the Pittsburgh Pirates after graduation. He played three seasons as a shortstop in the lower rungs of the Pirates farm system, first at Batavia in the New York-Penn League in 1962, then at Gastonia in the Western Carolinas League from 1963 to 1965 where he was an all-star shortstop. While in Gastonia, NC, Cook taught U. S. History at Hunter Huss High School, often leaving early in the day in-season for away games in North and South Carolina. He sometimes employed a baseball-modeled game in class to drill his students in historical facts. He retired as a player in 1965 to take over as general manager of the Gastonia Pirates, then was hired by the Pirates the next year to work in their front office. In 1968, he became Assistant Farm Director, then in 1976 was promoted to Director of Minor Leagues and Scouting.

On January 24, 1983, Murray Cook left the Pirates, for whom he had worked for 21 years, to join the New York Yankees as Director of Player Development. The Yankees front office was in turmoil in those years, as owner George Steinbrenner was continually firing his managers and general managers in the hope of returning to the World Series. 1982 had been a particularly trying year, as the Yankees finished fifth in the American League East with a 79-83 record, their first losing season since 1973, while burning through three managers. This instability also brought some opportunity, though, and for Cook it meant that he was promoted to general manager on June 30, 1983, taking over for Bill Bergesch, who had held the position since the end of 1980 although never having the formal title. Only George Selkirk had been a Canadian GM in MLB before Cook. In the usual confusion that surrounded the Yankee front office in those years, Bergesch and his assistant Cedric Tallis remained on the payroll, with senior titles but undefined responsibilities. With manager Billy Martin at the helm the whole season, the team showed a marked improvement, posting a 91-71 record, good for third place. However, Steinbrenner fired Martin for the third time (there would be two more repeats) after the season, and brought in Yogi Berra as the new manager; Cook had little say in these matters.

Murray Cook earned Steinbrenner's wrath when the Oakland Athletics selected pitcher Tim Belcher from the Yankees in the free agent compensation draft on February 8, 1984. Belcher had been the first player selected in the 1983 amateur draft but had refused to sign with the Minnesota Twins at the time; the Yankees had scored a coup by selecting him in the secondary phase of the January 1984 amateur draft and signing him to a contract a few days later. However, they had already submitted their list of protected players for the free agent draft and were unable to add the newly-signed Belcher, leading to his selection by the Athletics. A furious Steinbrenner reduced Cook's responsibilities, leaving a void in the front office until he named Clyde King as his replacement on April 9th. Indeed, player agents and rival executives had commented that they did not know anymore whom to call in the Yankee front office to discuss contracts or trades, while in the meantime Steinbrenner had entered into a public row with Berra over his line-up selection in the early days of the season. Cook inherited the title of Vice-President and Director of Scouting in the shuffle, as Steinbrenner stated that "I take the blame of asking a young man to move too fast too soon". He stayed in that role for a few months until he submitted his resignation on August 20th, explaining that he wanted to have a decision-making role again.

Murray Cook did not stay unemployed for long. On August 30th of that same year, the Montreal Expos' top brass fired manager Bill Virdon, followed five days later by general manager John McHale who was fired upstairs. The team had been steadily regressing since its breakthrough season in 1979, seeing its winning percentage fall slightly every year since that season, and had missed the playoffs the last two years in spite of great front-line talent. It was now clear that the team had to rebuild, and Cook was hired on September 5th with that mandate. His first action after taking over was to find a new manager, and on November 14th, he hired Buck Rodgers, who had just won the Minor League Manager of the Year Award by leading the Expos' AAA farm team, the Indianapolis Indians, to a 91-63 record and the American Association championship. Rodgers would prove to be an inspired choice, pushing the Expos to overachieve notably over the next few seasons.

The Expos' ownership was very concerned with the spiralling player salaries that were sweeping baseball in the mid-1980s and instructed Cook to operate within a tight budget. There were no more free agent signings, and a number of the team's biggest stars had to be moved before having a chance to leave the team via free agency. As a result, Cook became known for the inordinate number of trades he engineered - over two dozen in three years - including one major blockbuster each off-season:

In addition to all the trades, Cook was a pioneer in finding talent on the minor league scrap heap. In contrast with McHale, who would always look for "proven veterans", often over-the-hill, to fill holes on the bench or at the back end of the pitching staff, Cook would scour the minor leagues for players who had never been given a fair shot or who had become available cheaply through other circumstances; he would then send them to AAA to show whether they could provide anything that the club could use and, if not, would be released without having cost the club anything but a few weeks' salary. Pitchers Dennis Martinez and Carlos Perez were the two prize catches to come through that route. He also began using the Rule V Draft, landing pitcher Jeff Parrett for next to nothing in 1985. All of these tacks would be pursued by his successors. These changes meant that, of the team he inherited in September 1984, only two recognizable figures - Tim Raines and Tim Wallach - were left by the end of the 1987 season. Yet, all of this upheaval had a positive effect. The Expos halted their slide and returned to competitiveness under Cook's watch, even figuring in the pennant race until the last few weeks of the 1987 season, something that had not happened since 1983.

Unfortunately for Cook, by the time the 1987 pennant race was heating up, he was gone. The Expos called a hasty press conference on August 11th to announce that Cook had been fired and that Bill Stoneman, Vice-President for Baseball Administration (i.e. responsible for salaries and other financial matters) was taking over as general manager. Over the next few weeks, rumors began circulating that the firing was not for baseball reasons. In fact, Murray Cook had been dismissed for having an affair with Pamela Brochu, wife of Expos President and CEO Claude Brochu. Cook broke up with his wife of many years to marry his former boss's wife, but after that it became difficult for him to find other senior positions in baseball.

After the 1987 season, he was hired to replace Bill Bergesch as GM of the Cincinnati Reds. The club performed well in 1988, finishing in second place and winning three more games than the previous season. However, the Reds fell to fifth in 1989. While the turmoil surrounding manager Pete Rose being banned from baseball surely had an impact on the team's performance on the field, Cook was replaced by Bob Quinn after the season.

After a few years in the wilderness, Cook landed other baseball jobs as a scout with the Boston Red Sox from 2002 to 2006, then with the Detroit Tigers starting in 2007. Cook shares a name with another Murray Cook, who is Major League Baseball's Field and Facilities Coordinator; the two are not related.